*2.1.1 First case* μrz > 0

For *εr<sup>θ</sup>* >0, we have.

$$|\varepsilon\_{r,\theta\overline{f}}^{\rm TE} = |\varepsilon\_{r\theta}| \left( \mathbf{1} - \frac{\mathbf{1}}{|\varepsilon\_{r\theta}\,\mu\_{rz}|k\_0^2} \cdot \left(\frac{u\_{nm}'}{R}\right)^2 \right) = |\varepsilon\_{r\theta}| \left( \mathbf{1} - \left(\frac{f\_{c.m\mathbf{n}}^{\rm TE}}{f}\right)^2 \right) < 0, \text{ if } f < f\_{c.m\mathbf{n}}^{\rm TE} \tag{30}$$

And for *εr<sup>θ</sup>* <0, *εTE <sup>r</sup>*,*eff* is rewritten as

$$\left| \varepsilon\_{r,\theta\overline{f}}^{TE} = -|\left| \varepsilon\_{r\theta} \right| \left( \mathbf{1} + \frac{\mathbf{1}}{|\left| \varepsilon\_{r\theta} \right| \mu\_{rx} |k\_0^2} . \left( \frac{u\_{nm}'}{R} \right)^2 \right) < \mathbf{0}. \tag{31}$$

It can be seen that *μrz* >0 leads to *εTE <sup>r</sup>*,*eff* < 0 below the cutoff frequency whenever *εr<sup>θ</sup>* >0 or *εr<sup>θ</sup>* <0.
